LMFDB - Embedded newform 1045.2.a.f.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$8.34436701122$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.7281497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1 $ x^6 - 2x^5 - 5x^4 + 7x^3 + 6x^2 - 2*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.326248$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.73890 q^{2} -0.901323 q^{3} +1.02379 q^{4} -1.00000 q^{5} -1.56731 q^{6} +2.22757 q^{7} -1.69754 q^{8} -2.18762 q^{9} -1.73890 q^{10} -1.00000 q^{11} -0.922763 q^{12} -3.64023 q^{13} +3.87353 q^{14} +0.901323 q^{15} -4.99943 q^{16} +0.644551 q^{17} -3.80406 q^{18} -1.00000 q^{19} -1.02379 q^{20} -2.00776 q^{21} -1.73890 q^{22} -6.15983 q^{23} +1.53003 q^{24} +1.00000 q^{25} -6.33001 q^{26} +4.67572 q^{27} +2.28056 q^{28} -1.88204 q^{29} +1.56731 q^{30} -0.183675 q^{31} -5.29846 q^{32} +0.901323 q^{33} +1.12081 q^{34} -2.22757 q^{35} -2.23966 q^{36} -4.11428 q^{37} -1.73890 q^{38} +3.28102 q^{39} +1.69754 q^{40} -1.53371 q^{41} -3.49130 q^{42} -1.53577 q^{43} -1.02379 q^{44} +2.18762 q^{45} -10.7113 q^{46} +1.75837 q^{47} +4.50610 q^{48} -2.03793 q^{49} +1.73890 q^{50} -0.580948 q^{51} -3.72682 q^{52} -2.81491 q^{53} +8.13063 q^{54} +1.00000 q^{55} -3.78139 q^{56} +0.901323 q^{57} -3.27269 q^{58} -3.90068 q^{59} +0.922763 q^{60} +0.734057 q^{61} -0.319392 q^{62} -4.87307 q^{63} +0.785358 q^{64} +3.64023 q^{65} +1.56731 q^{66} -1.30264 q^{67} +0.659883 q^{68} +5.55199 q^{69} -3.87353 q^{70} +10.5493 q^{71} +3.71357 q^{72} -5.17599 q^{73} -7.15433 q^{74} -0.901323 q^{75} -1.02379 q^{76} -2.22757 q^{77} +5.70538 q^{78} +2.89974 q^{79} +4.99943 q^{80} +2.34852 q^{81} -2.66698 q^{82} +13.8209 q^{83} -2.05552 q^{84} -0.644551 q^{85} -2.67055 q^{86} +1.69633 q^{87} +1.69754 q^{88} -6.39884 q^{89} +3.80406 q^{90} -8.10886 q^{91} -6.30635 q^{92} +0.165550 q^{93} +3.05763 q^{94} +1.00000 q^{95} +4.77562 q^{96} +6.23352 q^{97} -3.54376 q^{98} +2.18762 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9} + 2 q^{10} - 6 q^{11} + q^{12} - 5 q^{13} - 8 q^{14} + q^{15} + 4 q^{16} + q^{17} + 6 q^{18} - 6 q^{19} - 4 q^{20} - 21 q^{21}+ \cdots - q^{99}+O(q^{100}) $ 6 * q - 2 * q^2 - q^3 + 4 * q^4 - 6 * q^5 + 5 * q^7 - 12 * q^8 + q^9 + 2 * q^10 - 6 * q^11 + q^12 - 5 * q^13 - 8 * q^14 + q^15 + 4 * q^16 + q^17 + 6 * q^18 - 6 * q^19 - 4 * q^20 - 21 * q^21 + 2 * q^22 + 4 * q^23 - q^24 + 6 * q^25 - 14 * q^26 - 16 * q^27 + 10 * q^28 - 9 * q^29 - 21 * q^31 - q^32 + q^33 - 5 * q^35 - 28 * q^36 - 3 * q^37 + 2 * q^38 + 20 * q^39 + 12 * q^40 - 23 * q^41 + q^42 + 7 * q^43 - 4 * q^44 - q^45 - 12 * q^46 - 18 * q^47 - 3 * q^49 - 2 * q^50 - 16 * q^51 + 13 * q^52 - 17 * q^53 + q^54 + 6 * q^55 - 2 * q^56 + q^57 + 23 * q^58 - 29 * q^59 - q^60 + 17 * q^61 + 2 * q^62 + 6 * q^63 - 18 * q^64 + 5 * q^65 + 8 * q^67 - q^68 - 38 * q^69 + 8 * q^70 - 12 * q^71 + 13 * q^72 + 2 * q^73 - 37 * q^74 - q^75 - 4 * q^76 - 5 * q^77 + q^78 + 3 * q^79 - 4 * q^80 - 2 * q^81 + 24 * q^82 - 11 * q^83 - 3 * q^84 - q^85 - 12 * q^86 - 12 * q^87 + 12 * q^88 - 22 * q^89 - 6 * q^90 - 18 * q^91 - 15 * q^92 + 18 * q^93 + 22 * q^94 + 6 * q^95 - 17 * q^96 - 2 * q^97 - q^98 - q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.73890	1.22959	0.614795	−	0.788687i	$-0.289239\pi$
$2$	1.73890	1.22959	0.614795	+	0.788687i	$0.289239\pi$
$3$	−0.901323	−0.520379	−0.260190	−	0.965558i	$-0.583785\pi$
$3$	−0.901323	−0.520379	−0.260190	+	0.965558i	$0.583785\pi$
$4$	1.02379	0.511894
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.56731	−0.639853
$7$	2.22757	0.841943	0.420971	−	0.907074i	$-0.361689\pi$
$7$	2.22757	0.841943	0.420971	+	0.907074i	$0.361689\pi$
$8$	−1.69754	−0.600171
$9$	−2.18762	−0.729206
$10$	−1.73890	−0.549890
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−0.922763	−0.266379
$13$	−3.64023	−1.00962	−0.504809	−	0.863231i	$-0.668437\pi$
$13$	−3.64023	−1.00962	−0.504809	+	0.863231i	$0.668437\pi$
$14$	3.87353	1.03525
$15$	0.901323	0.232721
$16$	−4.99943	−1.24986
$17$	0.644551	0.156327	0.0781633	−	0.996941i	$-0.475094\pi$
$17$	0.644551	0.156327	0.0781633	+	0.996941i	$0.475094\pi$
$18$	−3.80406	−0.896625
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.02379	−0.228926
$21$	−2.00776	−0.438129
$22$	−1.73890	−0.370736
$23$	−6.15983	−1.28441	−0.642206	−	0.766532i	$-0.721981\pi$
$23$	−6.15983	−1.28441	−0.642206	+	0.766532i	$0.721981\pi$
$24$	1.53003	0.312316
$25$	1.00000	0.200000
$26$	−6.33001	−1.24142
$27$	4.67572	0.899842
$28$	2.28056	0.430985
$29$	−1.88204	−0.349487	−0.174743	−	0.984614i	$-0.555910\pi$
$29$	−1.88204	−0.349487	−0.174743	+	0.984614i	$0.555910\pi$
$30$	1.56731	0.286151
$31$	−0.183675	−0.0329889	−0.0164945	−	0.999864i	$-0.505251\pi$
$31$	−0.183675	−0.0329889	−0.0164945	+	0.999864i	$0.505251\pi$
$32$	−5.29846	−0.936644
$33$	0.901323	0.156900
$34$	1.12081	0.192218
$35$	−2.22757	−0.376528
$36$	−2.23966	−0.373276
$37$	−4.11428	−0.676383	−0.338192	−	0.941077i	$-0.609815\pi$
$37$	−4.11428	−0.676383	−0.338192	+	0.941077i	$0.609815\pi$
$38$	−1.73890	−0.282088
$39$	3.28102	0.525384
$40$	1.69754	0.268405
$41$	−1.53371	−0.239525	−0.119763	−	0.992803i	$-0.538213\pi$
$41$	−1.53371	−0.239525	−0.119763	+	0.992803i	$0.538213\pi$
$42$	−3.49130	−0.538720
$43$	−1.53577	−0.234202	−0.117101	−	0.993120i	$-0.537360\pi$
$43$	−1.53577	−0.234202	−0.117101	+	0.993120i	$0.537360\pi$
$44$	−1.02379	−0.154342
$45$	2.18762	0.326111
$46$	−10.7113	−1.57930
$47$	1.75837	0.256484	0.128242	−	0.991743i	$-0.459067\pi$
$47$	1.75837	0.256484	0.128242	+	0.991743i	$0.459067\pi$
$48$	4.50610	0.650400
$49$	−2.03793	−0.291133
$50$	1.73890	0.245918
$51$	−0.580948	−0.0813491
$52$	−3.72682	−0.516817
$53$	−2.81491	−0.386658	−0.193329	−	0.981134i	$-0.561929\pi$
$53$	−2.81491	−0.386658	−0.193329	+	0.981134i	$0.561929\pi$
$54$	8.13063	1.10644
$55$	1.00000	0.134840
$56$	−3.78139	−0.505309
$57$	0.901323	0.119383
$58$	−3.27269	−0.429726
$59$	−3.90068	−0.507825	−0.253913	−	0.967227i	$-0.581718\pi$
$59$	−3.90068	−0.507825	−0.253913	+	0.967227i	$0.581718\pi$
$60$	0.922763	0.119128
$61$	0.734057	0.0939863	0.0469932	−	0.998895i	$-0.485036\pi$
$61$	0.734057	0.0939863	0.0469932	+	0.998895i	$0.485036\pi$
$62$	−0.319392	−0.0405629
$63$	−4.87307	−0.613949
$64$	0.785358	0.0981698
$65$	3.64023	0.451515
$66$	1.56731	0.192923
$67$	−1.30264	−0.159143	−0.0795717	−	0.996829i	$-0.525355\pi$
$67$	−1.30264	−0.159143	−0.0795717	+	0.996829i	$0.525355\pi$
$68$	0.659883	0.0800226
$69$	5.55199	0.668381
$70$	−3.87353	−0.462976
$71$	10.5493	1.25197	0.625983	−	0.779837i	$-0.284698\pi$
$71$	10.5493	1.25197	0.625983	+	0.779837i	$0.284698\pi$
$72$	3.71357	0.437648
$73$	−5.17599	−0.605804	−0.302902	−	0.953022i	$-0.597956\pi$
$73$	−5.17599	−0.605804	−0.302902	+	0.953022i	$0.597956\pi$
$74$	−7.15433	−0.831674
$75$	−0.901323	−0.104076
$76$	−1.02379	−0.117437
$77$	−2.22757	−0.253855
$78$	5.70538	0.646007
$79$	2.89974	0.326246	0.163123	−	0.986606i	$-0.447843\pi$
$79$	2.89974	0.326246	0.163123	+	0.986606i	$0.447843\pi$
$80$	4.99943	0.558954
$81$	2.34852	0.260947
$82$	−2.66698	−0.294518
$83$	13.8209	1.51704	0.758522	−	0.651647i	$-0.225922\pi$
$83$	13.8209	1.51704	0.758522	+	0.651647i	$0.225922\pi$
$84$	−2.05552	−0.224276
$85$	−0.644551	−0.0699114
$86$	−2.67055	−0.287973
$87$	1.69633	0.181866
$88$	1.69754	0.180958
$89$	−6.39884	−0.678275	−0.339138	−	0.940737i	$-0.610135\pi$
$89$	−6.39884	−0.678275	−0.339138	+	0.940737i	$0.610135\pi$
$90$	3.80406	0.400983
$91$	−8.10886	−0.850040
$92$	−6.30635	−0.657483
$93$	0.165550	0.0171667
$94$	3.05763	0.315371
$95$	1.00000	0.102598
$96$	4.77562	0.487410
$97$	6.23352	0.632918	0.316459	−	0.948606i	$-0.397506\pi$
$97$	6.23352	0.632918	0.316459	+	0.948606i	$0.397506\pi$
$98$	−3.54376	−0.357974
$99$	2.18762	0.219864
$100$	1.02379	0.102379
$101$	−16.3563	−1.62751	−0.813756	−	0.581206i	$-0.802581\pi$
$101$	−16.3563	−1.62751	−0.813756	+	0.581206i	$0.802581\pi$
$102$	−1.01021	−0.100026
$103$	0.645519	0.0636049	0.0318025	−	0.999494i	$-0.489875\pi$
$103$	0.645519	0.0636049	0.0318025	+	0.999494i	$0.489875\pi$
$104$	6.17943	0.605943
$105$	2.00776	0.195937
$106$	−4.89487	−0.475432
$107$	−17.5900	−1.70049	−0.850243	−	0.526390i	$-0.823545\pi$
$107$	−17.5900	−1.70049	−0.850243	+	0.526390i	$0.823545\pi$
$108$	4.78694	0.460624
$109$	19.4359	1.86162	0.930808	−	0.365507i	$-0.119104\pi$
$109$	19.4359	1.86162	0.930808	+	0.365507i	$0.119104\pi$
$110$	1.73890	0.165798
$111$	3.70829	0.351976
$112$	−11.1366	−1.05231
$113$	−5.88383	−0.553504	−0.276752	−	0.960941i	$-0.589258\pi$
$113$	−5.88383	−0.553504	−0.276752	+	0.960941i	$0.589258\pi$
$114$	1.56731	0.146792
$115$	6.15983	0.574407
$116$	−1.92681	−0.178900
$117$	7.96342	0.736219
$118$	−6.78291	−0.624417
$119$	1.43578	0.131618
$120$	−1.53003	−0.139672
$121$	1.00000	0.0909091
$122$	1.27645	0.115565
$123$	1.38237	0.124644
$124$	−0.188044	−0.0168868
$125$	−1.00000	−0.0894427
$126$	−8.47381	−0.754907
$127$	15.9676	1.41690	0.708450	−	0.705762i	$-0.249395\pi$
$127$	15.9676	1.41690	0.708450	+	0.705762i	$0.249395\pi$
$128$	11.9626	1.05735
$129$	1.38422	0.121874
$130$	6.33001	0.555178
$131$	−3.93173	−0.343517	−0.171759	−	0.985139i	$-0.554945\pi$
$131$	−3.93173	−0.343517	−0.171759	+	0.985139i	$0.554945\pi$
$132$	0.922763	0.0803162
$133$	−2.22757	−0.193155
$134$	−2.26517	−0.195681
$135$	−4.67572	−0.402422
$136$	−1.09415	−0.0938226
$137$	15.5422	1.32786	0.663929	−	0.747796i	$-0.268888\pi$
$137$	15.5422	1.32786	0.663929	+	0.747796i	$0.268888\pi$
$138$	9.65438	0.821836
$139$	−6.69900	−0.568202	−0.284101	−	0.958794i	$-0.591695\pi$
$139$	−6.69900	−0.568202	−0.284101	+	0.958794i	$0.591695\pi$
$140$	−2.28056	−0.192742
$141$	−1.58486	−0.133469
$142$	18.3441	1.53941
$143$	3.64023	0.304411
$144$	10.9368	0.911404
$145$	1.88204	0.156295
$146$	−9.00056	−0.744891
$147$	1.83683	0.151499
$148$	−4.21215	−0.346236
$149$	0.783045	0.0641496	0.0320748	−	0.999485i	$-0.489789\pi$
$149$	0.783045	0.0641496	0.0320748	+	0.999485i	$0.489789\pi$
$150$	−1.56731	−0.127971
$151$	8.46749	0.689075	0.344537	−	0.938773i	$-0.388036\pi$
$151$	8.46749	0.689075	0.344537	+	0.938773i	$0.388036\pi$
$152$	1.69754	0.137689
$153$	−1.41003	−0.113994
$154$	−3.87353	−0.312138
$155$	0.183675	0.0147531
$156$	3.35907	0.268941
$157$	8.27024	0.660037	0.330019	−	0.943974i	$-0.392945\pi$
$157$	8.27024	0.660037	0.330019	+	0.943974i	$0.392945\pi$
$158$	5.04237	0.401150
$159$	2.53715	0.201209
$160$	5.29846	0.418880
$161$	−13.7215	−1.08140
$162$	4.08385	0.320858
$163$	23.9030	1.87223	0.936113	−	0.351699i	$-0.114396\pi$
$163$	23.9030	1.87223	0.936113	+	0.351699i	$0.114396\pi$
$164$	−1.57019	−0.122612
$165$	−0.901323	−0.0701679
$166$	24.0333	1.86534
$167$	−19.4405	−1.50435	−0.752176	−	0.658962i	$-0.770996\pi$
$167$	−19.4405	−1.50435	−0.752176	+	0.658962i	$0.770996\pi$
$168$	3.40825	0.262952
$169$	0.251253	0.0193272
$170$	−1.12081	−0.0859624
$171$	2.18762	0.167291
$172$	−1.57230	−0.119887
$173$	−11.6888	−0.888682	−0.444341	−	0.895858i	$-0.646562\pi$
$173$	−11.6888	−0.888682	−0.444341	+	0.895858i	$0.646562\pi$
$174$	2.94975	0.223620
$175$	2.22757	0.168389
$176$	4.99943	0.376847
$177$	3.51577	0.264261
$178$	−11.1270	−0.834001
$179$	−6.77452	−0.506351	−0.253176	−	0.967420i	$-0.581475\pi$
$179$	−6.77452	−0.506351	−0.253176	+	0.967420i	$0.581475\pi$
$180$	2.23966	0.166934
$181$	−4.34551	−0.322999	−0.161500	−	0.986873i	$-0.551633\pi$
$181$	−4.34551	−0.322999	−0.161500	+	0.986873i	$0.551633\pi$
$182$	−14.1005	−1.04520
$183$	−0.661622	−0.0489085
$184$	10.4565	0.770867
$185$	4.11428	0.302488
$186$	0.287876	0.0211081
$187$	−0.644551	−0.0471342
$188$	1.80020	0.131293
$189$	10.4155	0.757616
$190$	1.73890	0.126153
$191$	−1.37612	−0.0995728	−0.0497864	−	0.998760i	$-0.515854\pi$
$191$	−1.37612	−0.0995728	−0.0497864	+	0.998760i	$0.515854\pi$
$192$	−0.707861	−0.0510855
$193$	5.15711	0.371217	0.185609	−	0.982624i	$-0.440574\pi$
$193$	5.15711	0.371217	0.185609	+	0.982624i	$0.440574\pi$
$194$	10.8395	0.778231
$195$	−3.28102	−0.234959
$196$	−2.08641	−0.149029
$197$	12.0998	0.862072	0.431036	−	0.902335i	$-0.358148\pi$
$197$	12.0998	0.862072	0.431036	+	0.902335i	$0.358148\pi$
$198$	3.80406	0.270343
$199$	−13.2948	−0.942445	−0.471223	−	0.882014i	$-0.656187\pi$
$199$	−13.2948	−0.942445	−0.471223	+	0.882014i	$0.656187\pi$
$200$	−1.69754	−0.120034
$201$	1.17410	0.0828149
$202$	−28.4420	−2.00117
$203$	−4.19239	−0.294248
$204$	−0.594768	−0.0416421
$205$	1.53371	0.107119
$206$	1.12250	0.0782080
$207$	13.4753	0.936601
$208$	18.1991	1.26188
$209$	1.00000	0.0691714
$210$	3.49130	0.240923
$211$	−26.0866	−1.79588	−0.897939	−	0.440120i	$-0.854936\pi$
$211$	−26.0866	−1.79588	−0.897939	+	0.440120i	$0.854936\pi$
$212$	−2.88188	−0.197928
$213$	−9.50828	−0.651497
$214$	−30.5873	−2.09090
$215$	1.53577	0.104738
$216$	−7.93722	−0.540059
$217$	−0.409148	−0.0277748
$218$	33.7971	2.28903
$219$	4.66524	0.315248
$220$	1.02379	0.0690238
$221$	−2.34631	−0.157830
$222$	6.44837	0.432786
$223$	13.8606	0.928175	0.464087	−	0.885789i	$-0.346382\pi$
$223$	13.8606	0.928175	0.464087	+	0.885789i	$0.346382\pi$
$224$	−11.8027	−0.788600
$225$	−2.18762	−0.145841
$226$	−10.2314	−0.680583
$227$	−12.9104	−0.856893	−0.428446	−	0.903567i	$-0.640939\pi$
$227$	−12.9104	−0.856893	−0.428446	+	0.903567i	$0.640939\pi$
$228$	0.922763	0.0611115
$229$	7.93710	0.524499	0.262249	−	0.965000i	$-0.415536\pi$
$229$	7.93710	0.524499	0.262249	+	0.965000i	$0.415536\pi$
$230$	10.7113	0.706285
$231$	2.00776	0.132101
$232$	3.19484	0.209752
$233$	17.6265	1.15475	0.577374	−	0.816480i	$-0.304078\pi$
$233$	17.6265	1.15475	0.577374	+	0.816480i	$0.304078\pi$
$234$	13.8476	0.905248
$235$	−1.75837	−0.114703
$236$	−3.99347	−0.259953
$237$	−2.61360	−0.169772
$238$	2.49669	0.161836
$239$	−18.6909	−1.20901	−0.604507	−	0.796600i	$-0.706630\pi$
$239$	−18.6909	−1.20901	−0.604507	+	0.796600i	$0.706630\pi$
$240$	−4.50610	−0.290868
$241$	−0.877285	−0.0565109	−0.0282555	−	0.999601i	$-0.508995\pi$
$241$	−0.877285	−0.0565109	−0.0282555	+	0.999601i	$0.508995\pi$
$242$	1.73890	0.111781
$243$	−16.1439	−1.03563
$244$	0.751518	0.0481110
$245$	2.03793	0.130198
$246$	2.40381	0.153261
$247$	3.64023	0.231622
$248$	0.311795	0.0197990
$249$	−12.4571	−0.789438
$250$	−1.73890	−0.109978
$251$	−0.448818	−0.0283291	−0.0141646	−	0.999900i	$-0.504509\pi$
$251$	−0.448818	−0.0283291	−0.0141646	+	0.999900i	$0.504509\pi$
$252$	−4.98899	−0.314277
$253$	6.15983	0.387265
$254$	27.7662	1.74221
$255$	0.580948	0.0363804
$256$	19.2311	1.20194
$257$	−7.85809	−0.490174	−0.245087	−	0.969501i	$-0.578817\pi$
$257$	−7.85809	−0.490174	−0.245087	+	0.969501i	$0.578817\pi$
$258$	2.40703	0.149855
$259$	−9.16484	−0.569476
$260$	3.72682	0.231128
$261$	4.11719	0.254848
$262$	−6.83691	−0.422386
$263$	0.585215	0.0360859	0.0180429	−	0.999837i	$-0.494256\pi$
$263$	0.585215	0.0360859	0.0180429	+	0.999837i	$0.494256\pi$
$264$	−1.53003	−0.0941669
$265$	2.81491	0.172919
$266$	−3.87353	−0.237502
$267$	5.76742	0.352960
$268$	−1.33363	−0.0814645
$269$	−3.59078	−0.218934	−0.109467	−	0.993990i	$-0.534914\pi$
$269$	−3.59078	−0.218934	−0.109467	+	0.993990i	$0.534914\pi$
$270$	−8.13063	−0.494814
$271$	−23.4466	−1.42428	−0.712139	−	0.702038i	$-0.752274\pi$
$271$	−23.4466	−1.42428	−0.712139	+	0.702038i	$0.752274\pi$
$272$	−3.22239	−0.195386
$273$	7.30870	0.442343
$274$	27.0264	1.63272
$275$	−1.00000	−0.0603023
$276$	5.68406	0.342140
$277$	−19.2912	−1.15909	−0.579547	−	0.814939i	$-0.696771\pi$
$277$	−19.2912	−1.15909	−0.579547	+	0.814939i	$0.696771\pi$
$278$	−11.6489	−0.698656
$279$	0.401809	0.0240557
$280$	3.78139	0.225981
$281$	−5.82550	−0.347520	−0.173760	−	0.984788i	$-0.555592\pi$
$281$	−5.82550	−0.347520	−0.173760	+	0.984788i	$0.555592\pi$
$282$	−2.75592	−0.164112
$283$	−15.5166	−0.922368	−0.461184	−	0.887305i	$-0.652575\pi$
$283$	−15.5166	−0.922368	−0.461184	+	0.887305i	$0.652575\pi$
$284$	10.8002	0.640874
$285$	−0.901323	−0.0533898
$286$	6.33001	0.374301
$287$	−3.41645	−0.201667
$288$	11.5910	0.683006
$289$	−16.5846	−0.975562
$290$	3.27269	0.192179
$291$	−5.61842	−0.329357
$292$	−5.29912	−0.310107
$293$	−12.8089	−0.748305	−0.374152	−	0.927367i	$-0.622066\pi$
$293$	−12.8089	−0.748305	−0.374152	+	0.927367i	$0.622066\pi$
$294$	3.19407	0.186282
$295$	3.90068	0.227106
$296$	6.98415	0.405945
$297$	−4.67572	−0.271313
$298$	1.36164	0.0788777
$299$	22.4232	1.29677
$300$	−0.922763	−0.0532758
$301$	−3.42103	−0.197185
$302$	14.7242	0.847280
$303$	14.7423	0.846923
$304$	4.99943	0.286737
$305$	−0.734057	−0.0420320
$306$	−2.45191	−0.140166
$307$	−14.0052	−0.799321	−0.399661	−	0.916663i	$-0.630872\pi$
$307$	−14.0052	−0.799321	−0.399661	+	0.916663i	$0.630872\pi$
$308$	−2.28056	−0.129947
$309$	−0.581821	−0.0330987
$310$	0.319392	0.0181403
$311$	4.10325	0.232674	0.116337	−	0.993210i	$-0.462885\pi$
$311$	4.10325	0.232674	0.116337	+	0.993210i	$0.462885\pi$
$312$	−5.56966	−0.315320
$313$	21.5394	1.21748	0.608740	−	0.793370i	$-0.291675\pi$
$313$	21.5394	1.21748	0.608740	+	0.793370i	$0.291675\pi$
$314$	14.3812	0.811576
$315$	4.87307	0.274567
$316$	2.96872	0.167004
$317$	−8.36917	−0.470059	−0.235030	−	0.971988i	$-0.575519\pi$
$317$	−8.36917	−0.470059	−0.235030	+	0.971988i	$0.575519\pi$
$318$	4.41186	0.247405
$319$	1.88204	0.105374
$320$	−0.785358	−0.0439029
$321$	15.8542	0.884898
$322$	−23.8603	−1.32968
$323$	−0.644551	−0.0358638
$324$	2.40439	0.133577
$325$	−3.64023	−0.201923
$326$	41.5650	2.30207
$327$	−17.5180	−0.968746
$328$	2.60353	0.143756
$329$	3.91689	0.215945
$330$	−1.56731	−0.0862778
$331$	−29.1194	−1.60055	−0.800274	−	0.599635i	$-0.795312\pi$
$331$	−29.1194	−1.60055	−0.800274	+	0.599635i	$0.795312\pi$
$332$	14.1497	0.776565
$333$	9.00046	0.493222
$334$	−33.8052	−1.84974
$335$	1.30264	0.0711711
$336$	10.0377	0.547600
$337$	3.36844	0.183491	0.0917453	−	0.995783i	$-0.470755\pi$
$337$	3.36844	0.183491	0.0917453	+	0.995783i	$0.470755\pi$
$338$	0.436905	0.0237645
$339$	5.30323	0.288032
$340$	−0.659883	−0.0357872
$341$	0.183675	0.00994653
$342$	3.80406	0.205700
$343$	−20.1326	−1.08706
$344$	2.60702	0.140561
$345$	−5.55199	−0.298909
$346$	−20.3257	−1.09272
$347$	3.97272	0.213267	0.106633	−	0.994298i	$-0.465993\pi$
$347$	3.97272	0.213267	0.106633	+	0.994298i	$0.465993\pi$
$348$	1.73668	0.0930959
$349$	1.51616	0.0811581	0.0405790	−	0.999176i	$-0.487080\pi$
$349$	1.51616	0.0811581	0.0405790	+	0.999176i	$0.487080\pi$
$350$	3.87353	0.207049
$351$	−17.0207	−0.908496
$352$	5.29846	0.282409
$353$	4.76876	0.253816	0.126908	−	0.991915i	$-0.459495\pi$
$353$	4.76876	0.253816	0.126908	+	0.991915i	$0.459495\pi$
$354$	6.11359	0.324934
$355$	−10.5493	−0.559896
$356$	−6.55105	−0.347205
$357$	−1.29410	−0.0684912
$358$	−11.7802	−0.622605
$359$	0.542750	0.0286453	0.0143226	−	0.999897i	$-0.495441\pi$
$359$	0.542750	0.0286453	0.0143226	+	0.999897i	$0.495441\pi$
$360$	−3.71357	−0.195722
$361$	1.00000	0.0526316
$362$	−7.55643	−0.397157
$363$	−0.901323	−0.0473072
$364$	−8.30176	−0.435130
$365$	5.17599	0.270924
$366$	−1.15050	−0.0601375
$367$	−26.1792	−1.36654	−0.683271	−	0.730165i	$-0.739443\pi$
$367$	−26.1792	−1.36654	−0.683271	+	0.730165i	$0.739443\pi$
$368$	30.7956	1.60533
$369$	3.35517	0.174663
$370$	7.15433	0.371936
$371$	−6.27042	−0.325544
$372$	0.169488	0.00878755
$373$	−12.7807	−0.661760	−0.330880	−	0.943673i	$-0.607345\pi$
$373$	−12.7807	−0.661760	−0.330880	+	0.943673i	$0.607345\pi$
$374$	−1.12081	−0.0579558
$375$	0.901323	0.0465441
$376$	−2.98490	−0.153934
$377$	6.85107	0.352848
$378$	18.1115	0.931557
$379$	−26.0931	−1.34031	−0.670157	−	0.742219i	$-0.733773\pi$
$379$	−26.0931	−1.34031	−0.670157	+	0.742219i	$0.733773\pi$
$380$	1.02379	0.0525192
$381$	−14.3920	−0.737325
$382$	−2.39295	−0.122434
$383$	−12.7812	−0.653091	−0.326545	−	0.945182i	$-0.605885\pi$
$383$	−12.7812	−0.653091	−0.326545	+	0.945182i	$0.605885\pi$
$384$	−10.7821	−0.550224
$385$	2.22757	0.113528
$386$	8.96773	0.456445
$387$	3.35967	0.170781
$388$	6.38180	0.323987
$389$	−8.97979	−0.455293	−0.227647	−	0.973744i	$-0.573103\pi$
$389$	−8.97979	−0.455293	−0.227647	+	0.973744i	$0.573103\pi$
$390$	−5.70538	−0.288903
$391$	−3.97032	−0.200788
$392$	3.45946	0.174729
$393$	3.54376	0.178759
$394$	21.0403	1.06000
$395$	−2.89974	−0.145902
$396$	2.23966	0.112547
$397$	26.8417	1.34715	0.673573	−	0.739120i	$-0.264758\pi$
$397$	26.8417	1.34715	0.673573	+	0.739120i	$0.264758\pi$
$398$	−23.1184	−1.15882
$399$	2.00776	0.100514
$400$	−4.99943	−0.249972
$401$	−9.25518	−0.462182	−0.231091	−	0.972932i	$-0.574229\pi$
$401$	−9.25518	−0.462182	−0.231091	+	0.972932i	$0.574229\pi$
$402$	2.04165	0.101828
$403$	0.668617	0.0333062
$404$	−16.7454	−0.833114
$405$	−2.34852	−0.116699
$406$	−7.29016	−0.361804
$407$	4.11428	0.203937
$408$	0.986183	0.0488233
$409$	26.0835	1.28975	0.644874	−	0.764289i	$-0.276910\pi$
$409$	26.0835	1.28975	0.644874	+	0.764289i	$0.276910\pi$
$410$	2.66698	0.131713
$411$	−14.0085	−0.690990
$412$	0.660875	0.0325590
$413$	−8.68904	−0.427560
$414$	23.4323	1.15164
$415$	−13.8209	−0.678443
$416$	19.2876	0.945652
$417$	6.03796	0.295680
$418$	1.73890	0.0850526
$419$	−32.5181	−1.58861	−0.794307	−	0.607517i	$-0.792166\pi$
$419$	−32.5181	−1.58861	−0.794307	+	0.607517i	$0.792166\pi$
$420$	2.05552	0.100299
$421$	3.16934	0.154464	0.0772320	−	0.997013i	$-0.475392\pi$
$421$	3.16934	0.154464	0.0772320	+	0.997013i	$0.475392\pi$
$422$	−45.3621	−2.20819
$423$	−3.84664	−0.187030
$424$	4.77843	0.232061
$425$	0.644551	0.0312653
$426$	−16.5340	−0.801074
$427$	1.63516	0.0791311
$428$	−18.0084	−0.870469
$429$	−3.28102	−0.158409
$430$	2.67055	0.128785
$431$	−1.19700	−0.0576575	−0.0288288	−	0.999584i	$-0.509178\pi$
$431$	−1.19700	−0.0576575	−0.0288288	+	0.999584i	$0.509178\pi$
$432$	−23.3759	−1.12468
$433$	23.4218	1.12558	0.562789	−	0.826601i	$-0.309728\pi$
$433$	23.4218	1.12558	0.562789	+	0.826601i	$0.309728\pi$
$434$	−0.711469	−0.0341516
$435$	−1.69633	−0.0813327
$436$	19.8982	0.952950
$437$	6.15983	0.294664
$438$	8.11241	0.387626
$439$	−21.4961	−1.02595	−0.512975	−	0.858403i	$-0.671457\pi$
$439$	−21.4961	−1.02595	−0.512975	+	0.858403i	$0.671457\pi$
$440$	−1.69754	−0.0809270
$441$	4.45821	0.212295
$442$	−4.08001	−0.194066
$443$	23.6133	1.12190	0.560950	−	0.827850i	$-0.310436\pi$
$443$	23.6133	1.12190	0.560950	+	0.827850i	$0.310436\pi$
$444$	3.79650	0.180174
$445$	6.39884	0.303334
$446$	24.1023	1.14128
$447$	−0.705777	−0.0333821
$448$	1.74944	0.0826533
$449$	−5.35004	−0.252484	−0.126242	−	0.991999i	$-0.540292\pi$
$449$	−5.35004	−0.252484	−0.126242	+	0.991999i	$0.540292\pi$
$450$	−3.80406	−0.179325
$451$	1.53371	0.0722196
$452$	−6.02379	−0.283335
$453$	−7.63194	−0.358580
$454$	−22.4499	−1.05363
$455$	8.10886	0.380149
$456$	−1.53003	−0.0716503
$457$	39.7929	1.86143	0.930716	−	0.365742i	$-0.119185\pi$
$457$	39.7929	1.86143	0.930716	+	0.365742i	$0.119185\pi$
$458$	13.8019	0.644919
$459$	3.01374	0.140669
$460$	6.30635	0.294035
$461$	−3.21566	−0.149768	−0.0748840	−	0.997192i	$-0.523859\pi$
$461$	−3.21566	−0.149768	−0.0748840	+	0.997192i	$0.523859\pi$
$462$	3.49130	0.162430
$463$	16.7065	0.776416	0.388208	−	0.921572i	$-0.373094\pi$
$463$	16.7065	0.776416	0.388208	+	0.921572i	$0.373094\pi$
$464$	9.40915	0.436809
$465$	−0.165550	−0.00767720
$466$	30.6507	1.41987
$467$	−37.8565	−1.75179	−0.875894	−	0.482503i	$-0.839728\pi$
$467$	−37.8565	−1.75179	−0.875894	+	0.482503i	$0.839728\pi$
$468$	8.15285	0.376866
$469$	−2.90173	−0.133990
$470$	−3.05763	−0.141038
$471$	−7.45416	−0.343469
$472$	6.62156	0.304782
$473$	1.53577	0.0706146
$474$	−4.54481	−0.208750
$475$	−1.00000	−0.0458831
$476$	1.46994	0.0673744
$477$	6.15796	0.281953
$478$	−32.5017	−1.48659
$479$	−16.6101	−0.758936	−0.379468	−	0.925205i	$-0.623893\pi$
$479$	−16.6101	−0.758936	−0.379468	+	0.925205i	$0.623893\pi$
$480$	−4.77562	−0.217976
$481$	14.9769	0.682888
$482$	−1.52552	−0.0694853
$483$	12.3675	0.562739
$484$	1.02379	0.0465358
$485$	−6.23352	−0.283050
$486$	−28.0727	−1.27341
$487$	29.2975	1.32760	0.663798	−	0.747912i	$-0.268944\pi$
$487$	29.2975	1.32760	0.663798	+	0.747912i	$0.268944\pi$
$488$	−1.24609	−0.0564079
$489$	−21.5443	−0.974267
$490$	3.54376	0.160091
$491$	−0.806717	−0.0364066	−0.0182033	−	0.999834i	$-0.505795\pi$
$491$	−0.806717	−0.0364066	−0.0182033	+	0.999834i	$0.505795\pi$
$492$	1.41525	0.0638045
$493$	−1.21307	−0.0546340
$494$	6.33001	0.284800
$495$	−2.18762	−0.0983261
$496$	0.918269	0.0412315
$497$	23.4992	1.05408
$498$	−21.6617	−0.970686
$499$	−30.7212	−1.37527	−0.687634	−	0.726058i	$-0.741351\pi$
$499$	−30.7212	−1.37527	−0.687634	+	0.726058i	$0.741351\pi$
$500$	−1.02379	−0.0457852
$501$	17.5222	0.782833
$502$	−0.780451	−0.0348332
$503$	−13.0527	−0.581991	−0.290995	−	0.956724i	$-0.593986\pi$
$503$	−13.0527	−0.581991	−0.290995	+	0.956724i	$0.593986\pi$
$504$	8.27223	0.368475
$505$	16.3563	0.727846
$506$	10.7113	0.476178
$507$	−0.226460	−0.0100574
$508$	16.3475	0.725302
$509$	−22.6615	−1.00445	−0.502226	−	0.864736i	$-0.667485\pi$
$509$	−22.6615	−1.00445	−0.502226	+	0.864736i	$0.667485\pi$
$510$	1.01021	0.0447330
$511$	−11.5299	−0.510052
$512$	9.51581	0.420544
$513$	−4.67572	−0.206438
$514$	−13.6645	−0.602713
$515$	−0.645519	−0.0284450
$516$	1.41715	0.0623865
$517$	−1.75837	−0.0773329
$518$	−15.9368	−0.700222
$519$	10.5354	0.462451
$520$	−6.17943	−0.270986
$521$	19.0608	0.835070	0.417535	−	0.908661i	$-0.362894\pi$
$521$	19.0608	0.835070	0.417535	+	0.908661i	$0.362894\pi$
$522$	7.15940	0.313358
$523$	15.1864	0.664054	0.332027	−	0.943270i	$-0.392268\pi$
$523$	15.1864	0.664054	0.332027	+	0.943270i	$0.392268\pi$
$524$	−4.02526	−0.175844
$525$	−2.00776	−0.0876259
$526$	1.01763	0.0443709
$527$	−0.118388	−0.00515704
$528$	−4.50610	−0.196103
$529$	14.9435	0.649716
$530$	4.89487	0.212619
$531$	8.53319	0.370309
$532$	−2.28056	−0.0988748
$533$	5.58305	0.241829
$534$	10.0290	0.433997
$535$	17.5900	0.760481
$536$	2.21129	0.0955132
$537$	6.10603	0.263495
$538$	−6.24402	−0.269199
$539$	2.03793	0.0877798
$540$	−4.78694	−0.205997
$541$	41.1432	1.76888	0.884442	−	0.466650i	$-0.154539\pi$
$541$	41.1432	1.76888	0.884442	+	0.466650i	$0.154539\pi$
$542$	−40.7714	−1.75128
$543$	3.91671	0.168082
$544$	−3.41513	−0.146422
$545$	−19.4359	−0.832540
$546$	12.7091	0.543901
$547$	40.7714	1.74326	0.871630	−	0.490164i	$-0.163063\pi$
$547$	40.7714	1.74326	0.871630	+	0.490164i	$0.163063\pi$
$548$	15.9119	0.679722
$549$	−1.60583	−0.0685354
$550$	−1.73890	−0.0741471
$551$	1.88204	0.0801778
$552$	−9.42473	−0.401143
$553$	6.45938	0.274681
$554$	−33.5455	−1.42521
$555$	−3.70829	−0.157408
$556$	−6.85835	−0.290859
$557$	−14.2315	−0.603009	−0.301504	−	0.953465i	$-0.597489\pi$
$557$	−14.2315	−0.603009	−0.301504	+	0.953465i	$0.597489\pi$
$558$	0.698708	0.0295787
$559$	5.59054	0.236454
$560$	11.1366	0.470607
$561$	0.580948	0.0245277
$562$	−10.1300	−0.427308
$563$	−11.1511	−0.469963	−0.234982	−	0.972000i	$-0.575503\pi$
$563$	−11.1511	−0.469963	−0.234982	+	0.972000i	$0.575503\pi$
$564$	−1.62256	−0.0683220
$565$	5.88383	0.247534
$566$	−26.9819	−1.13414
$567$	5.23149	0.219702
$568$	−17.9078	−0.751393
$569$	11.4022	0.478005	0.239003	−	0.971019i	$-0.423180\pi$
$569$	11.4022	0.478005	0.239003	+	0.971019i	$0.423180\pi$
$570$	−1.56731	−0.0656476
$571$	12.5111	0.523573	0.261787	−	0.965126i	$-0.415688\pi$
$571$	12.5111	0.523573	0.261787	+	0.965126i	$0.415688\pi$
$572$	3.72682	0.155826
$573$	1.24033	0.0518156
$574$	−5.94088	−0.247967
$575$	−6.15983	−0.256883
$576$	−1.71806	−0.0715860
$577$	7.95520	0.331179	0.165590	−	0.986195i	$-0.447047\pi$
$577$	7.95520	0.331179	0.165590	+	0.986195i	$0.447047\pi$
$578$	−28.8390	−1.19954
$579$	−4.64822	−0.193174
$580$	1.92681	0.0800066
$581$	30.7871	1.27726
$582$	−9.76989	−0.404975
$583$	2.81491	0.116582
$584$	8.78645	0.363586
$585$	−7.96342	−0.329247
$586$	−22.2735	−0.920109
$587$	45.7680	1.88905	0.944524	−	0.328443i	$-0.106524\pi$
$587$	45.7680	1.88905	0.944524	+	0.328443i	$0.106524\pi$
$588$	1.88052	0.0775515
$589$	0.183675	0.00756818
$590$	6.78291	0.279248
$591$	−10.9058	−0.448604
$592$	20.5691	0.845383
$593$	−26.3613	−1.08253	−0.541264	−	0.840853i	$-0.682054\pi$
$593$	−26.3613	−1.08253	−0.541264	+	0.840853i	$0.682054\pi$
$594$	−8.13063	−0.333604
$595$	−1.43578	−0.0588614
$596$	0.801672	0.0328378
$597$	11.9829	0.490429
$598$	38.9917	1.59449
$599$	17.3368	0.708361	0.354180	−	0.935177i	$-0.384760\pi$
$599$	17.3368	0.708361	0.354180	+	0.935177i	$0.384760\pi$
$600$	1.53003	0.0624633
$601$	−15.7946	−0.644274	−0.322137	−	0.946693i	$-0.604401\pi$
$601$	−15.7946	−0.644274	−0.322137	+	0.946693i	$0.604401\pi$
$602$	−5.94884	−0.242457
$603$	2.84969	0.116048
$604$	8.66891	0.352733
$605$	−1.00000	−0.0406558
$606$	25.6355	1.04137
$607$	−1.80754	−0.0733659	−0.0366829	−	0.999327i	$-0.511679\pi$
$607$	−1.80754	−0.0733659	−0.0366829	+	0.999327i	$0.511679\pi$
$608$	5.29846	0.214881
$609$	3.77869	0.153120
$610$	−1.27645	−0.0516821
$611$	−6.40086	−0.258951
$612$	−1.44357	−0.0583529
$613$	10.9246	0.441239	0.220619	−	0.975360i	$-0.429192\pi$
$613$	10.9246	0.441239	0.220619	+	0.975360i	$0.429192\pi$
$614$	−24.3538	−0.982838
$615$	−1.38237	−0.0557425
$616$	3.78139	0.152357
$617$	−2.68742	−0.108191	−0.0540957	−	0.998536i	$-0.517228\pi$
$617$	−2.68742	−0.108191	−0.0540957	+	0.998536i	$0.517228\pi$
$618$	−1.01173	−0.0406978
$619$	18.2439	0.733284	0.366642	−	0.930362i	$-0.380507\pi$
$619$	18.2439	0.733284	0.366642	+	0.930362i	$0.380507\pi$
$620$	0.188044	0.00755202
$621$	−28.8016	−1.15577
$622$	7.13515	0.286094
$623$	−14.2539	−0.571069
$624$	−16.4032	−0.656655
$625$	1.00000	0.0400000
$626$	37.4550	1.49700
$627$	−0.901323	−0.0359954
$628$	8.46697	0.337869
$629$	−2.65186	−0.105737
$630$	8.47381	0.337604
$631$	9.12012	0.363066	0.181533	−	0.983385i	$-0.441894\pi$
$631$	9.12012	0.363066	0.181533	+	0.983385i	$0.441894\pi$
$632$	−4.92243	−0.195804
$633$	23.5125	0.934537
$634$	−14.5532	−0.577981
$635$	−15.9676	−0.633657
$636$	2.59750	0.102998
$637$	7.41852	0.293932
$638$	3.27269	0.129567
$639$	−23.0777	−0.912940
$640$	−11.9626	−0.472862
$641$	−2.90773	−0.114848	−0.0574241	−	0.998350i	$-0.518289\pi$
$641$	−2.90773	−0.114848	−0.0574241	+	0.998350i	$0.518289\pi$
$642$	27.5690	1.08806
$643$	−3.35603	−0.132349	−0.0661745	−	0.997808i	$-0.521079\pi$
$643$	−3.35603	−0.132349	−0.0661745	+	0.997808i	$0.521079\pi$
$644$	−14.0479	−0.553563
$645$	−1.38422	−0.0545036
$646$	−1.12081	−0.0440978
$647$	26.5428	1.04351	0.521753	−	0.853096i	$-0.325278\pi$
$647$	26.5428	1.04351	0.521753	+	0.853096i	$0.325278\pi$
$648$	−3.98670	−0.156613
$649$	3.90068	0.153115
$650$	−6.33001	−0.248283
$651$	0.368774	0.0144534
$652$	24.4716	0.958381
$653$	28.6844	1.12251	0.561253	−	0.827644i	$-0.310319\pi$
$653$	28.6844	1.12251	0.561253	+	0.827644i	$0.310319\pi$
$654$	−30.4621	−1.19116
$655$	3.93173	0.153626
$656$	7.66768	0.299373
$657$	11.3231	0.441756
$658$	6.81110	0.265524
$659$	−39.0719	−1.52203	−0.761013	−	0.648737i	$-0.775297\pi$
$659$	−39.0719	−1.52203	−0.761013	+	0.648737i	$0.775297\pi$
$660$	−0.922763	−0.0359185
$661$	−21.1901	−0.824199	−0.412099	−	0.911139i	$-0.635204\pi$
$661$	−21.1901	−0.824199	−0.412099	+	0.911139i	$0.635204\pi$
$662$	−50.6358	−1.96802
$663$	2.11478	0.0821314
$664$	−23.4616	−0.910486
$665$	2.22757	0.0863815
$666$	15.6509	0.606462
$667$	11.5931	0.448885
$668$	−19.9030	−0.770069
$669$	−12.4929	−0.483003
$670$	2.26517	0.0875113
$671$	−0.734057	−0.0283379
$672$	10.6380	0.410371
$673$	−24.6931	−0.951849	−0.475924	−	0.879486i	$-0.657886\pi$
$673$	−24.6931	−0.951849	−0.475924	+	0.879486i	$0.657886\pi$
$674$	5.85739	0.225618
$675$	4.67572	0.179968
$676$	0.257230	0.00989345
$677$	5.41516	0.208122	0.104061	−	0.994571i	$-0.466816\pi$
$677$	5.41516	0.208122	0.104061	+	0.994571i	$0.466816\pi$
$678$	9.22180	0.354161
$679$	13.8856	0.532881
$680$	1.09415	0.0419588
$681$	11.6364	0.445909
$682$	0.319392	0.0122302
$683$	−13.7885	−0.527604	−0.263802	−	0.964577i	$-0.584977\pi$
$683$	−13.7885	−0.527604	−0.263802	+	0.964577i	$0.584977\pi$
$684$	2.23966	0.0856354
$685$	−15.5422	−0.593836
$686$	−35.0087	−1.33664
$687$	−7.15389	−0.272938
$688$	7.67796	0.292719
$689$	10.2469	0.390377
$690$	−9.65438	−0.367536
$691$	−31.5750	−1.20117	−0.600586	−	0.799560i	$-0.705066\pi$
$691$	−31.5750	−1.20117	−0.600586	+	0.799560i	$0.705066\pi$
$692$	−11.9668	−0.454911
$693$	4.87307	0.185113
$694$	6.90818	0.262231
$695$	6.69900	0.254108
$696$	−2.87959	−0.109150
$697$	−0.988554	−0.0374442
$698$	2.63645	0.0997912
$699$	−15.8871	−0.600906
$700$	2.28056	0.0861971
$701$	−17.4459	−0.658924	−0.329462	−	0.944169i	$-0.606867\pi$
$701$	−17.4459	−0.658924	−0.329462	+	0.944169i	$0.606867\pi$
$702$	−29.5973	−1.11708
$703$	4.11428	0.155173
$704$	−0.785358	−0.0295993
$705$	1.58486	0.0596892
$706$	8.29242	0.312089
$707$	−36.4348	−1.37027
$708$	3.59940	0.135274
$709$	−38.7266	−1.45441	−0.727205	−	0.686421i	$-0.759181\pi$
$709$	−38.7266	−1.45441	−0.727205	+	0.686421i	$0.759181\pi$
$710$	−18.3441	−0.688443
$711$	−6.34352	−0.237901
$712$	10.8623	0.407081
$713$	1.13140	0.0423714
$714$	−2.25032	−0.0842162
$715$	−3.64023	−0.136137
$716$	−6.93567	−0.259198
$717$	16.8465	0.629145
$718$	0.943791	0.0352220
$719$	−28.9402	−1.07929	−0.539644	−	0.841894i	$-0.681441\pi$
$719$	−28.9402	−1.07929	−0.539644	+	0.841894i	$0.681441\pi$
$720$	−10.9368	−0.407592
$721$	1.43794	0.0535517
$722$	1.73890	0.0647153
$723$	0.790717	0.0294071
$724$	−4.44888	−0.165341
$725$	−1.88204	−0.0698973
$726$	−1.56731	−0.0581685
$727$	−0.331738	−0.0123035	−0.00615174	−	0.999981i	$-0.501958\pi$
$727$	−0.331738	−0.0123035	−0.00615174	+	0.999981i	$0.501958\pi$
$728$	13.7651	0.510169
$729$	7.50533	0.277975
$730$	9.00056	0.333126
$731$	−0.989879	−0.0366120
$732$	−0.677361	−0.0250360
$733$	−30.5241	−1.12743	−0.563716	−	0.825969i	$-0.690629\pi$
$733$	−30.5241	−1.12743	−0.563716	+	0.825969i	$0.690629\pi$
$734$	−45.5231	−1.68029
$735$	−1.83683	−0.0677525
$736$	32.6376	1.20304
$737$	1.30264	0.0479835
$738$	5.83432	0.214764
$739$	−8.65314	−0.318311	−0.159155	−	0.987254i	$-0.550877\pi$
$739$	−8.65314	−0.318311	−0.159155	+	0.987254i	$0.550877\pi$
$740$	4.21215	0.154842
$741$	−3.28102	−0.120531
$742$	−10.9037	−0.400286
$743$	33.9919	1.24704	0.623521	−	0.781807i	$-0.285702\pi$
$743$	33.9919	1.24704	0.623521	+	0.781807i	$0.285702\pi$
$744$	−0.281028	−0.0103030
$745$	−0.783045	−0.0286886
$746$	−22.2244	−0.813694
$747$	−30.2349	−1.10624
$748$	−0.659883	−0.0241277
$749$	−39.1829	−1.43171
$750$	1.56731	0.0572302
$751$	−50.2581	−1.83394	−0.916972	−	0.398951i	$-0.869374\pi$
$751$	−50.2581	−1.83394	−0.916972	+	0.398951i	$0.869374\pi$
$752$	−8.79085	−0.320569
$753$	0.404530	0.0147419
$754$	11.9133	0.433859
$755$	−8.46749	−0.308164
$756$	10.6633	0.387819
$757$	10.7948	0.392344	0.196172	−	0.980570i	$-0.437149\pi$
$757$	10.7948	0.392344	0.196172	+	0.980570i	$0.437149\pi$
$758$	−45.3734	−1.64804
$759$	−5.55199	−0.201525
$760$	−1.69754	−0.0615762
$761$	15.5661	0.564272	0.282136	−	0.959374i	$-0.408957\pi$
$761$	15.5661	0.564272	0.282136	+	0.959374i	$0.408957\pi$
$762$	−25.0263	−0.906608
$763$	43.2947	1.56737
$764$	−1.40886	−0.0509707
$765$	1.41003	0.0509798
$766$	−22.2254	−0.803035
$767$	14.1994	0.512709
$768$	−17.3334	−0.625465
$769$	19.6520	0.708671	0.354335	−	0.935118i	$-0.384707\pi$
$769$	19.6520	0.708671	0.354335	+	0.935118i	$0.384707\pi$
$770$	3.87353	0.139592
$771$	7.08267	0.255076
$772$	5.27979	0.190024
$773$	−41.0208	−1.47541	−0.737707	−	0.675121i	$-0.764092\pi$
$773$	−41.0208	−1.47541	−0.737707	+	0.675121i	$0.764092\pi$
$774$	5.84214	0.209991
$775$	−0.183675	−0.00659778
$776$	−10.5817	−0.379859
$777$	8.26048	0.296343
$778$	−15.6150	−0.559825
$779$	1.53371	0.0549509
$780$	−3.35907	−0.120274
$781$	−10.5493	−0.377482
$782$	−6.90401	−0.246887
$783$	−8.79991	−0.314483
$784$	10.1885	0.363874
$785$	−8.27024	−0.295178
$786$	6.16226	0.219801
$787$	25.2114	0.898688	0.449344	−	0.893359i	$-0.351658\pi$
$787$	25.2114	0.898688	0.449344	+	0.893359i	$0.351658\pi$
$788$	12.3876	0.441289
$789$	−0.527467	−0.0187783
$790$	−5.04237	−0.179400
$791$	−13.1066	−0.466018
$792$	−3.71357	−0.131956
$793$	−2.67213	−0.0948902
$794$	46.6752	1.65644
$795$	−2.53715	−0.0899833
$796$	−13.6111	−0.482432
$797$	−23.4053	−0.829057	−0.414528	−	0.910036i	$-0.636053\pi$
$797$	−23.4053	−0.829057	−0.414528	+	0.910036i	$0.636053\pi$
$798$	3.49130	0.123591
$799$	1.13336	0.0400953
$800$	−5.29846	−0.187329
$801$	13.9982	0.494602
$802$	−16.0939	−0.568294
$803$	5.17599	0.182657
$804$	1.20203	0.0423924
$805$	13.7215	0.483618
$806$	1.16266	0.0409530
$807$	3.23645	0.113929
$808$	27.7655	0.976786
$809$	−25.3235	−0.890325	−0.445163	−	0.895450i	$-0.646854\pi$
$809$	−25.3235	−0.890325	−0.445163	+	0.895450i	$0.646854\pi$
$810$	−4.08385	−0.143492
$811$	−1.24610	−0.0437566	−0.0218783	−	0.999761i	$-0.506965\pi$
$811$	−1.24610	−0.0437566	−0.0218783	+	0.999761i	$0.506965\pi$
$812$	−4.29211	−0.150624
$813$	21.1329	0.741165
$814$	7.15433	0.250759
$815$	−23.9030	−0.837285
$816$	2.90441	0.101675
$817$	1.53577	0.0537296
$818$	45.3567	1.58586
$819$	17.7391	0.619854
$820$	1.57019	0.0548335
$821$	20.4113	0.712359	0.356180	−	0.934418i	$-0.384079\pi$
$821$	20.4113	0.712359	0.356180	+	0.934418i	$0.384079\pi$
$822$	−24.3595	−0.849634
$823$	−15.4655	−0.539093	−0.269546	−	0.962987i	$-0.586874\pi$
$823$	−15.4655	−0.539093	−0.269546	+	0.962987i	$0.586874\pi$
$824$	−1.09579	−0.0381738
$825$	0.901323	0.0313800
$826$	−15.1094	−0.525723
$827$	−46.2008	−1.60656	−0.803280	−	0.595602i	$-0.796913\pi$
$827$	−46.2008	−1.60656	−0.803280	+	0.595602i	$0.796913\pi$
$828$	13.7959	0.479440
$829$	2.91356	0.101192	0.0505961	−	0.998719i	$-0.483888\pi$
$829$	2.91356	0.101192	0.0505961	+	0.998719i	$0.483888\pi$
$830$	−24.0333	−0.834207
$831$	17.3876	0.603168
$832$	−2.85888	−0.0991139
$833$	−1.31355	−0.0455117
$834$	10.4994	0.363566
$835$	19.4405	0.672767
$836$	1.02379	0.0354084
$837$	−0.858810	−0.0296848
$838$	−56.5459	−1.95334
$839$	28.2887	0.976633	0.488317	−	0.872667i	$-0.337611\pi$
$839$	28.2887	0.976633	0.488317	+	0.872667i	$0.337611\pi$
$840$	−3.40825	−0.117596
$841$	−25.4579	−0.877859
$842$	5.51117	0.189928
$843$	5.25066	0.180842
$844$	−26.7072	−0.919299
$845$	−0.251253	−0.00864337
$846$	−6.68893	−0.229970
$847$	2.22757	0.0765402
$848$	14.0730	0.483268
$849$	13.9855	0.479981
$850$	1.12081	0.0384435
$851$	25.3432	0.868755
$852$	−9.73446	−0.333497
$853$	41.9234	1.43543	0.717715	−	0.696337i	$-0.245188\pi$
$853$	41.9234	1.43543	0.717715	+	0.696337i	$0.245188\pi$
$854$	2.84339	0.0972989
$855$	−2.18762	−0.0748149
$856$	29.8597	1.02058
$857$	52.4352	1.79115	0.895576	−	0.444909i	$-0.146764\pi$
$857$	52.4352	1.79115	0.895576	+	0.444909i	$0.146764\pi$
$858$	−5.70538	−0.194778
$859$	−21.2305	−0.724374	−0.362187	−	0.932105i	$-0.617970\pi$
$859$	−21.2305	−0.724374	−0.362187	+	0.932105i	$0.617970\pi$
$860$	1.57230	0.0536149
$861$	3.07932	0.104943
$862$	−2.08147	−0.0708952
$863$	1.75582	0.0597690	0.0298845	−	0.999553i	$-0.490486\pi$
$863$	1.75582	0.0597690	0.0298845	+	0.999553i	$0.490486\pi$
$864$	−24.7741	−0.842832
$865$	11.6888	0.397431
$866$	40.7282	1.38400
$867$	14.9480	0.507662
$868$	−0.418881	−0.0142177
$869$	−2.89974	−0.0983670
$870$	−2.94975	−0.100006
$871$	4.74192	0.160674
$872$	−32.9931	−1.11729
$873$	−13.6366	−0.461528
$874$	10.7113	0.362317
$875$	−2.22757	−0.0753056
$876$	4.77622	0.161373
$877$	27.7873	0.938309	0.469155	−	0.883116i	$-0.344559\pi$
$877$	27.7873	0.938309	0.469155	+	0.883116i	$0.344559\pi$
$878$	−37.3796	−1.26150
$879$	11.5450	0.389402
$880$	−4.99943	−0.168531
$881$	38.8712	1.30960	0.654802	−	0.755800i	$-0.272752\pi$
$881$	38.8712	1.30960	0.654802	+	0.755800i	$0.272752\pi$
$882$	7.75239	0.261037
$883$	−1.20467	−0.0405404	−0.0202702	−	0.999795i	$-0.506453\pi$
$883$	−1.20467	−0.0405404	−0.0202702	+	0.999795i	$0.506453\pi$
$884$	−2.40212	−0.0807922
$885$	−3.51577	−0.118181
$886$	41.0612	1.37948
$887$	−44.0947	−1.48056	−0.740278	−	0.672301i	$-0.765306\pi$
$887$	−44.0947	−1.48056	−0.740278	+	0.672301i	$0.765306\pi$
$888$	−6.29497	−0.211245
$889$	35.5691	1.19295
$890$	11.1270	0.372977
$891$	−2.34852	−0.0786784
$892$	14.1903	0.475127
$893$	−1.75837	−0.0588415
$894$	−1.22728	−0.0410463
$895$	6.77452	0.226447
$896$	26.6475	0.890230
$897$	−20.2105	−0.674809
$898$	−9.30320	−0.310452
$899$	0.345683	0.0115292
$900$	−2.23966	−0.0746552
$901$	−1.81436	−0.0604450
$902$	2.66698	0.0888006
$903$	3.08345	0.102611
$904$	9.98803	0.332197
$905$	4.34551	0.144450
$906$	−13.2712	−0.440907
$907$	36.2297	1.20299	0.601493	−	0.798878i	$-0.294573\pi$
$907$	36.2297	1.20299	0.601493	+	0.798878i	$0.294573\pi$
$908$	−13.2175	−0.438638
$909$	35.7813	1.18679
$910$	14.1005	0.467428
$911$	−5.41417	−0.179380	−0.0896898	−	0.995970i	$-0.528588\pi$
$911$	−5.41417	−0.179380	−0.0896898	+	0.995970i	$0.528588\pi$
$912$	−4.50610	−0.149212
$913$	−13.8209	−0.457406
$914$	69.1960	2.28880
$915$	0.661622	0.0218726
$916$	8.12591	0.268488
$917$	−8.75822	−0.289222
$918$	5.24060	0.172966
$919$	54.0931	1.78437	0.892183	−	0.451675i	$-0.149173\pi$
$919$	54.0931	1.78437	0.892183	+	0.451675i	$0.149173\pi$
$920$	−10.4565	−0.344742
$921$	12.6232	0.415950
$922$	−5.59172	−0.184153
$923$	−38.4017	−1.26401
$924$	2.05552	0.0676217
$925$	−4.11428	−0.135277
$926$	29.0510	0.954674
$927$	−1.41215	−0.0463811
$928$	9.97193	0.327345
$929$	52.0785	1.70864	0.854320	−	0.519747i	$-0.173974\pi$
$929$	52.0785	1.70864	0.854320	+	0.519747i	$0.173974\pi$
$930$	−0.287876	−0.00943981
$931$	2.03793	0.0667904
$932$	18.0457	0.591108
$933$	−3.69835	−0.121079
$934$	−65.8288	−2.15398
$935$	0.644551	0.0210791
$936$	−13.5182	−0.441857
$937$	35.9861	1.17561	0.587807	−	0.809002i	$-0.299992\pi$
$937$	35.9861	1.17561	0.587807	+	0.809002i	$0.299992\pi$
$938$	−5.04584	−0.164752
$939$	−19.4140	−0.633551
$940$	−1.80020	−0.0587159
$941$	11.9972	0.391098	0.195549	−	0.980694i	$-0.437351\pi$
$941$	11.9972	0.391098	0.195549	+	0.980694i	$0.437351\pi$
$942$	−12.9621	−0.422327
$943$	9.44739	0.307649
$944$	19.5012	0.634709
$945$	−10.4155	−0.338816
$946$	2.67055	0.0868271
$947$	37.1663	1.20774	0.603871	−	0.797082i	$-0.293624\pi$
$947$	37.1663	1.20774	0.603871	+	0.797082i	$0.293624\pi$
$948$	−2.67577	−0.0869051
$949$	18.8418	0.611630
$950$	−1.73890	−0.0564175
$951$	7.54332	0.244609
$952$	−2.43730	−0.0789933
$953$	16.5500	0.536106	0.268053	−	0.963404i	$-0.413620\pi$
$953$	16.5500	0.536106	0.268053	+	0.963404i	$0.413620\pi$
$954$	10.7081	0.346687
$955$	1.37612	0.0445303
$956$	−19.1355	−0.618886
$957$	−1.69633	−0.0548345
$958$	−28.8834	−0.933180
$959$	34.6213	1.11798
$960$	0.707861	0.0228461
$961$	−30.9663	−0.998912
$962$	26.0434	0.839673
$963$	38.4801	1.24000
$964$	−0.898154	−0.0289276
$965$	−5.15711	−0.166013
$966$	21.5058	0.691939
$967$	37.7443	1.21377	0.606887	−	0.794788i	$-0.292418\pi$
$967$	37.7443	1.21377	0.606887	+	0.794788i	$0.292418\pi$
$968$	−1.69754	−0.0545610
$969$	0.580948	0.0186628
$970$	−10.8395	−0.348035
$971$	−40.2912	−1.29301	−0.646504	−	0.762911i	$-0.723769\pi$
$971$	−40.2912	−1.29301	−0.646504	+	0.762911i	$0.723769\pi$
$972$	−16.5280	−0.530134
$973$	−14.9225	−0.478393
$974$	50.9455	1.63240
$975$	3.28102	0.105077
$976$	−3.66987	−0.117470
$977$	1.24031	0.0396812	0.0198406	−	0.999803i	$-0.493684\pi$
$977$	1.24031	0.0396812	0.0198406	+	0.999803i	$0.493684\pi$
$978$	−37.4635	−1.19795
$979$	6.39884	0.204508
$980$	2.08641	0.0666478
$981$	−42.5182	−1.35750
$982$	−1.40280	−0.0447653
$983$	8.75471	0.279232	0.139616	−	0.990206i	$-0.455413\pi$
$983$	8.75471	0.279232	0.139616	+	0.990206i	$0.455413\pi$
$984$	−2.34662	−0.0748077
$985$	−12.0998	−0.385530
$986$	−2.10942	−0.0671775
$987$	−3.53038	−0.112373
$988$	3.72682	0.118566
$989$	9.46005	0.300812
$990$	−3.80406	−0.120901
$991$	49.7066	1.57898	0.789491	−	0.613763i	$-0.210345\pi$
$991$	49.7066	1.57898	0.789491	+	0.613763i	$0.210345\pi$
$992$	0.973192	0.0308989
$993$	26.2460	0.832891
$994$	40.8629	1.29609
$995$	13.2948	0.421474
$996$	−12.7534	−0.404108
$997$	46.3005	1.46635	0.733176	−	0.680039i	$-0.238037\pi$
$997$	46.3005	1.46635	0.733176	+	0.680039i	$0.238037\pi$
$998$	−53.4211	−1.69102
$999$	−19.2372	−0.608638

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.f.1.6	✓	6
3.2	odd	2		9405.2.a.z.1.1		6
5.4	even	2		5225.2.a.l.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.f.1.6	✓	6	1.1	even	1	trivial
5225.2.a.l.1.1		6	5.4	even	2
9405.2.a.z.1.1		6	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73890 q^{2} -0.901323 q^{3} +1.02379 q^{4} -1.00000 q^{5} -1.56731 q^{6} +2.22757 q^{7} -1.69754 q^{8} -2.18762 q^{9} -1.73890 q^{10} -1.00000 q^{11} -0.922763 q^{12} -3.64023 q^{13} +3.87353 q^{14} +0.901323 q^{15} -4.99943 q^{16} +0.644551 q^{17} -3.80406 q^{18} -1.00000 q^{19} -1.02379 q^{20} -2.00776 q^{21} -1.73890 q^{22} -6.15983 q^{23} +1.53003 q^{24} +1.00000 q^{25} -6.33001 q^{26} +4.67572 q^{27} +2.28056 q^{28} -1.88204 q^{29} +1.56731 q^{30} -0.183675 q^{31} -5.29846 q^{32} +0.901323 q^{33} +1.12081 q^{34} -2.22757 q^{35} -2.23966 q^{36} -4.11428 q^{37} -1.73890 q^{38} +3.28102 q^{39} +1.69754 q^{40} -1.53371 q^{41} -3.49130 q^{42} -1.53577 q^{43} -1.02379 q^{44} +2.18762 q^{45} -10.7113 q^{46} +1.75837 q^{47} +4.50610 q^{48} -2.03793 q^{49} +1.73890 q^{50} -0.580948 q^{51} -3.72682 q^{52} -2.81491 q^{53} +8.13063 q^{54} +1.00000 q^{55} -3.78139 q^{56} +0.901323 q^{57} -3.27269 q^{58} -3.90068 q^{59} +0.922763 q^{60} +0.734057 q^{61} -0.319392 q^{62} -4.87307 q^{63} +0.785358 q^{64} +3.64023 q^{65} +1.56731 q^{66} -1.30264 q^{67} +0.659883 q^{68} +5.55199 q^{69} -3.87353 q^{70} +10.5493 q^{71} +3.71357 q^{72} -5.17599 q^{73} -7.15433 q^{74} -0.901323 q^{75} -1.02379 q^{76} -2.22757 q^{77} +5.70538 q^{78} +2.89974 q^{79} +4.99943 q^{80} +2.34852 q^{81} -2.66698 q^{82} +13.8209 q^{83} -2.05552 q^{84} -0.644551 q^{85} -2.67055 q^{86} +1.69633 q^{87} +1.69754 q^{88} -6.39884 q^{89} +3.80406 q^{90} -8.10886 q^{91} -6.30635 q^{92} +0.165550 q^{93} +3.05763 q^{94} +1.00000 q^{95} +4.77562 q^{96} +6.23352 q^{97} -3.54376 q^{98} +2.18762 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9} + 2 q^{10} - 6 q^{11} + q^{12} - 5 q^{13} - 8 q^{14} + q^{15} + 4 q^{16} + q^{17} + 6 q^{18} - 6 q^{19} - 4 q^{20} - 21 q^{21}+ \cdots - q^{99}+O(q^{100}) \) 6 * q - 2 * q^2 - q^3 + 4 * q^4 - 6 * q^5 + 5 * q^7 - 12 * q^8 + q^9 + 2 * q^10 - 6 * q^11 + q^12 - 5 * q^13 - 8 * q^14 + q^15 + 4 * q^16 + q^17 + 6 * q^18 - 6 * q^19 - 4 * q^20 - 21 * q^21 + 2 * q^22 + 4 * q^23 - q^24 + 6 * q^25 - 14 * q^26 - 16 * q^27 + 10 * q^28 - 9 * q^29 - 21 * q^31 - q^32 + q^33 - 5 * q^35 - 28 * q^36 - 3 * q^37 + 2 * q^38 + 20 * q^39 + 12 * q^40 - 23 * q^41 + q^42 + 7 * q^43 - 4 * q^44 - q^45 - 12 * q^46 - 18 * q^47 - 3 * q^49 - 2 * q^50 - 16 * q^51 + 13 * q^52 - 17 * q^53 + q^54 + 6 * q^55 - 2 * q^56 + q^57 + 23 * q^58 - 29 * q^59 - q^60 + 17 * q^61 + 2 * q^62 + 6 * q^63 - 18 * q^64 + 5 * q^65 + 8 * q^67 - q^68 - 38 * q^69 + 8 * q^70 - 12 * q^71 + 13 * q^72 + 2 * q^73 - 37 * q^74 - q^75 - 4 * q^76 - 5 * q^77 + q^78 + 3 * q^79 - 4 * q^80 - 2 * q^81 + 24 * q^82 - 11 * q^83 - 3 * q^84 - q^85 - 12 * q^86 - 12 * q^87 + 12 * q^88 - 22 * q^89 - 6 * q^90 - 18 * q^91 - 15 * q^92 + 18 * q^93 + 22 * q^94 + 6 * q^95 - 17 * q^96 - 2 * q^97 - q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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